Expected Cost
|
+- [f]
|    0
|
+- [Program]
|    0:While(z ≥ 1 + y)
|        Tick(1)
|        y :~ {1 : 1 + y}
|    1:While(y ≥ 3)
|        Tick(1)
|        y :~ {1 : -3 + y}
|
+- Expected Cost
|  |
|  +- [f]
|  |    0
|  |
|  +- [Program]
|  |    0:While(z ≥ 1 + y)
|  |        Tick(1)
|  |        y :~ {1 : 1 + y}
|  |
|  +- While.step
|  |  |
|  |  +- [Problem]
|  |  |    0:While(z ≥ 1 + y)
|  |  |        Tick(1)
|  |  |        y :~ {1 : 1 + y}
|  |  |
|  |  +- [f]
|  |  |    0
|  |  |
|  |  +- Expected Cost Body
|  |  |  |
|  |  |  `- [1 | ⊤]
|  |  |
|  |  +- linear-template
|  |  |  |
|  |  |  `- 1 + [-y + z | -y + z ≥ 0]
|  |  |
|  |  +- [Norms]
|  |  |    [[1 | ⊤],[-y + z | -y + z ≥ 0]]
|  |  |
|  |  +- [Invariant]
|  |  |    z ≥ 1 + y ==> [1 | ⊤] + h([1 | ⊤],[-1 + -y + z | -1 + -y + z ≥ 0]) ≼ h([1 | ⊤],[-y + z | -y + z ≥ 0])
|  |  |    1 + y ≥ 1 + z ==> 0 ≼ h([1 | ⊤],[-y + z | -y + z ≥ 0])
|  |  |
|  |  `- [-y + z | -y + z ≥ 0]
|  |
|  `- [-y + z | -y + z ≥ 0]
|
+- Expected Cost
|  |
|  +- [f]
|  |    0
|  |
|  +- [Program]
|  |    1:While(y ≥ 3)
|  |        Tick(1)
|  |        y :~ {1 : -3 + y}
|  |
|  +- While.step
|  |  |
|  |  +- [Problem]
|  |  |    1:While(y ≥ 3)
|  |  |        Tick(1)
|  |  |        y :~ {1 : -3 + y}
|  |  |
|  |  +- [f]
|  |  |    0
|  |  |
|  |  +- Expected Cost Body
|  |  |  |
|  |  |  `- [1 | ⊤]
|  |  |
|  |  +- linear-template
|  |  |  |
|  |  |  `- 1 + [-2 + y | -2 + y ≥ 0]
|  |  |
|  |  +- [Norms]
|  |  |    [[1 | ⊤],[-2 + y | -2 + y ≥ 0]]
|  |  |
|  |  +- [Invariant]
|  |  |    y ≥ 3 ==> [1 | ⊤] + h([1 | ⊤],[-5 + y | -5 + y ≥ 0]) ≼ h([1 | ⊤],[-2 + y | -2 + y ≥ 0])
|  |  |    3 ≥ 1 + y ==> 0 ≼ h([1 | ⊤],[-2 + y | -2 + y ≥ 0])
|  |  |
|  |  `- [-2 + y | -2 + y ≥ 0]
|  |
|  `- [-2 + y | -2 + y ≥ 0]
|
+- Expected Cost
|  |
|  +- [f]
|  |    [-2 + y | -2 + y ≥ 0]
|  |
|  +- [Program]
|  |    0:While(z ≥ 1 + y)
|  |        Tick(1)
|  |        y :~ {1 : 1 + y}
|  |
|  +- While.step
|  |  |
|  |  +- [Problem]
|  |  |    0:While(z ≥ 1 + y)
|  |  |        Tick(1)
|  |  |        y :~ {1 : 1 + y}
|  |  |
|  |  +- [f]
|  |  |    [-2 + y | -2 + y ≥ 0]
|  |  |
|  |  +- linear-template
|  |  |  |
|  |  |  `- 1 + [-2 + y | -2 + y ≥ 0] + 2·([-y + z | -y + z ≥ 0])
|  |  |
|  |  +- [Norms]
|  |  |    [[1 | ⊤],[-2 + y | -2 + y ≥ 0],[-y + z | -y + z ≥ 0]]
|  |  |
|  |  +- [Invariant]
|  |  |    z ≥ 1 + y ==> 0 + h([1 | ⊤],[-1 + y | -1 + y ≥ 0],[-1 + -y + z | -1 + -y + z ≥ 0]) ≼ h([1 | ⊤],[-2 + y | -2 + y ≥ 0],[-y + z | -y + z ≥ 0])
|  |  |    1 + y ≥ 1 + z ==> [-2 + y | -2 + y ≥ 0] ≼ h([1 | ⊤],[-2 + y | -2 + y ≥ 0],[-y + z | -y + z ≥ 0])
|  |  |
|  |  `- [-2 + y | -2 + y ≥ 0] + [-y + z | -y + z ≥ 0]
|  |
|  `- [-2 + y | -2 + y ≥ 0] + [-y + z | -y + z ≥ 0]
|
`- [-y + z | -y + z ≥ 0] + [-2 + y | -2 + y ≥ 0] + [-y + z | -y + z ≥ 0]